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ECSE304_sol2

# ECSE304_sol2 - ECSE 304B 2010 Homework Assignment 2...

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ECSE 304B 2010 Homework Assignment 2 Solution Set 1. Signal Spaces

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2. Properties of Signal Spaces (a) Consider a set of delayed sinc functions ( 29 ( 29 { } sinc t nT T - for n = 0, ±1, ±2, . .. where sinc( t ) = sin( π t )/( π t ). These functions are mutually orthogonal, yet overlap in time and frequency domains. (Note that these sinc functions appear in the sampling theorem.) (b) Pairwise orthogonal => ( ), ( ) 0 i j x t x t = for any i j , i = 1, …, n and j = 1, …, n . So, 2 1 1 1 1 1 1 2 1 2 1 2 2 2 1 2 2 1 1 ( ( ) ( )) ( ) ( ),( ( ) ( )) ( ), ( ) ( ), ( ) ( ), ( ) ( ), ( ) ( ), ( ) ( ), ( ) ( ), ( ) ( ), ( ) ( ), ( ) ( ), ( ) ( ) n n n n n n n n n n n i i i i i x t x t dt x t x t x t x t x t x t x t x t x t x t x t x t x t x t x t x t x t x t x t x t x t x t x t x t x t = = + + = + + + + = + + + + + + + + + + + = = (c) Since ( ) y t V , 1 1 ( ) ( ), ( ) ( ) ( ) i i i i i i y t y t b t b t y b t = = = = . So,

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ECSE304_sol2 - ECSE 304B 2010 Homework Assignment 2...

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